Modeling and Control of an Ornithopter for Non-Equilibrium Maneuvers by Cameron Jarrel Rose A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering - Electrical Engineering and Computer Sciences in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Ronald S. Fearing, Chair Professor Pieter Abbeel Professor Robert Dudley Fall 2015 Modeling and Control of an Ornithopter for Non-Equilibrium Maneuvers Copyright 2015 by Cameron Jarrel Rose 1 Abstract Modeling and Control of an Ornithopter for Non-Equilibrium Maneuvers by Cameron Jarrel Rose Doctor of Philosophy in Engineering - Electrical Engineering and Computer Sciences University of California, Berkeley Professor Ronald S. Fearing, Chair Flapping-winged flight is very complex, and it is difficult to efficiently model the un- steady airflow and nonlinear dynamics for online control. While steady state flight is well understood, transitions between flight regimes are not readily modeled or controlled. Ma- neuverability in non-equilibrium flight, which birds and insects readily exhibit in nature, is necessary to operate in the types of cluttered environments that small-scale flapping-winged robots are best suited for. The advantages of flapping wings over quadrotors and fixed-wing fliers are realized in the ability to transition from forward flight to hover to gliding. Flight in the transitions between these regimes necessitates the development of novel modeling techniques and online control techniques to accurately complete these types of maneuvers. In this thesis, methods for modeling and controlling the transitions between takeoff and diving maneuvers are developed for a flapping-winged micro aerial vehicle (MAV), the H2Bird. To transition into takeoff and steady state flight, a cooperative launching system is developed for the H2Bird by carrying it on the back of a 32 gram hexapedal millirobot, the VelociRoACH. The necessary initial velocity and pitch angle are determined for take off using force data collected in a wind tunnel, and the VelociRoACH is used to reach these initial conditions for successful launch. The models for the diving maneuver are generated using an automatic piece-wise affine identification technique. The flight conditions during the maneuver are segmented into separate regions and least-squares is used to estimate affine linear models for each modeling region. These models are used to compute the reachability sets for the recovery conditions for safe diving, and linear quadratic regulator controllers are used to maintain stable conditions before and after the dive. The data-driven automatic modeling techniques and controller design processes can be extended to additional flight maneuvers. i To my grandfather For supporting me with words of wisdom and encouragement along the way, and always letting me know how proud of me he was. It meant a lot to me. ii Contents Contents ii List of Figures iv List of Tables viii 1 Introduction and Background 1 1.1 Introduction . 1 1.2 Background . 4 2 Hardware and Robotic Platform 6 2.1 Ornithopter Platform . 6 2.2 Aerodynamics . 7 3 Comparison of Wind Tunnel Force Measurements with Free Flight 10 3.1 Flight Data Collection . 10 3.2 Comparison of Data Sets . 14 4 Takeoff and Flight Transition 21 4.1 Robotic Platforms and Behaviors . 21 4.2 Control and Launching . 23 4.3 Experimental Results and Discussion . 24 5 Piece-wise Linear Modeling for Diving 35 5.1 Piece-wise Affine Modeling . 35 5.2 Reachability Analysis . 41 6 Online Control for Diving 44 6.1 Control Implementation . 44 6.2 Experiments and Discussion . 48 7 Conclusions 53 iii A Reachability Sets 55 A.1 Position Reachability Sets . 55 A.2 Velocity and Pitch Angle Reachability Sets . 58 Bibliography 60 iv List of Figures 1.1 The launch sequence of the H2Bird from a cradle on the back of the VelociRoACH. 2 1.2 The H2Bird dive sequence. Label 1 indicates the point at which the robot tran- sitions from level flight to the unpowered dive, label 2 indicates the transition between the unpowered dive to the powered recovery, label 3 indicates the lowest point in the dive and the transition between the recovery back to level flight, and label 4 indicates flight to a new height. 2 2.1 The H2Bird ornithopter [17]. 6 2.2 The free body diagram for the wind tunnel data [4]. 7 2.3 H2Bird mounted to sensor . 8 2.4 Aerodynamic horizontal force surface in world coordinates. 9 2.5 Aerodynamic vertical force surface in world coordinates. 9 2.6 Aerodynamic pitch moment surface. 9 2.7 Thrust (blue) and lift (green) forces as a function of the duty cycle. 9 2.8 Pitch moment as a function of the duty cycle. 9 3.1 Block diagram of the H2Bird control system. 11 3.2 Diagram of the H2Bird mounted to the force-torque sensor in the wind tunnel [36]. 12 3.3 Free body diagram of the H2Bird for the wind tunnel experiment data. 13 3.4 A sample plot of the vertical force surface measured in the wind tunnel as a function of angle of attack and wind speed. 14 3.5 The range of the change in pitch moment that the elevator can achieve for 80 percent duty cycle for different angles of attack and wind speeds. For example, at 40◦ angle of attack and 2.5 m/s wind speed, the elevator can affect a maximum change of about 1.1 N*m pitch moment through its entire range. 15 3.6 Side and top views of a sample flight path with start point (green square) and stop point (red circle) in the tracking space. The black bar indicates the target path. 16 3.7 The pitch, pitch velocity, elevator input, and velocity magnitude of the H2Bird during one trial. 17 3.8 Block diagram of the estimation of equilibrium points from the wind tunnel data. 18 v 3.9 Equilibrium points measured in free flight (red squares) and equilibrium points predicted from the wind tunnel (blue triangles). 19 4.1 Free body diagram for the H2Bird. 22 4.2 Net lift over a range of angles of attack and wind speeds at 16 Hz flap speed. The dashed black line indicates the line of zero net vertical force. Above the line are feasible conditions for takeoff and infeasible conditions are below. 22 4.3 VelociRoACH with cradle and H2Bird ornithopter MAV (top), and launch se- quence from left to right (bottom). 23 4.4 The launch cradle on top of the VelociRoACH, highlighted by the red dashed line. The carbon fiber spars through the back of the cradle are shown in the top-left inset. 24 4.5 Telemetry data at the start of running for a single launch trial. 25 4.6 Telemetry data around launch for a single launch trial. The red line indicates the launch point. 26 4.7 Launch experiments for varied running speeds and launch angles. The shaded region represented the wind tunnel predicted failure area, and the unshaded region represents the predicted success area. The red double triangles represent failures in the predicted success region. 27 4.8 The change in pitch angle 0.2 seconds post-launch for each tested velocity (left) and the elevator input at launch for each tested velocity (right). The red double triangles represent failures in the predicted success region. 28 4.9 The change in pitch angle vs. the elevator input for each trial. The red double triangles represent failures in the predicted success region. 29 4.10 Telemetry data for a single trial for the VelociRoACH running alone (left) and running with an inertial mass equivalent to the H2Bird(right). 30 4.11 Telemetry data for a single trial for the VelociRoACH running with the pas- sive H2Bird (left) and running with the H2Bird flapping at 5 Hz (right). 31 4.12 Roll velocity variance (left) and pitch velocity variance (right) for the VelociRoACH alone, with inertial mass, with passive H2Bird, and with active H2Bird. 32 4.13 Average running velocities (left) and average power consumed (right) at steady state for VelociRoACH alone, with inertial mass, with passive H2Bird, and with active H2Bird. 32 5.1 Free body diagram of the H2Bird for the relevant state variables for the discrete- time models [4]. 36 5.2 Position data for a single trial of the open loop H2Bird diving experiment. The gray shading represents the unpowered portion of the dive. The black marker indicates the conditions at the lowest vertical position in the dive. 37 vi 5.3 Telemetry data for a single trial of the open-loop H2Birddiving experiment over the time period. The position is on the left and the horizontal velocity, vertical velocity, and pitch angle are on the right. The gray shading represents the un- powered portion of the dive. The black marker indicates the conditions at the lowest vertical position in the dive. 37 5.4 The means of the conditions for the lowest point in the open loop dive trials. The error bars represent one standard deviation above and below the mean for each state variable. 38 5.5 Graphical representation of the K-means segmentation of the dive (left) and es- cape (right) portions of the open loop experiments in the horizontal velocity, vertical velocity, and pitch angle space. 39 5.6 The terminal polytopes for the backwards reachability analysis. The set in the horizontal and vertical position space is on the left, and the set in the horizontal velocity, vertical velocity, and pitch angle space is on the right. The goal position is always zero and all other heights are relative to the goal.